The collineation groups of the translation planes of order 25

Czerwinski and Oakden have classified all the translation planes of order 25. In this paper the author gives the order and orbit lengths on the line at infinity of the collineation groups of these planes.     

Supersolvable Frobenius groups with nilpotent centralizers

Let FH be a supersolvable Frobenius group with kernel F and complement H. Suppose that a finite group G admits FH as a group of automorphisms in such a manner that $C_G(F)=1$ and $C_G(H)$ is nilpotent of class c. We show that G is nilpotent of $(c,\left|FH\right|)$-bounded class.     

Reducible subgroups of exceptional algebraic groups

Let G be a simple algebraic group over an algebraically closed field. A closed subgroup H of G is called G-completely reducible (G-cr) if, whenever H is contained in a parabolic subgroup P of G, it is contained in a Levi factor of P. In this paper we complete the classification of connected G-cr subgroups when G has exceptional type, by determining the $L_0$-irreducible connected reductive subgroups for each simple classical factor $L_0$ of a Levi subgroup of G. As an illustration, we determine all reducible, G-cr semisimple subgroups when G has type $F_4$ and various properties thereof. This work complements results of Lawther, Liebeck, Seitz and Testerman, and is vital in classifying non-G-cr reductive subgroups, a project being undertaken by the authors elsewhere.     

On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002     

Finite orbit modules for parabolic subgroups of exceptional groups

Let G be a reductive linear algebraic group, P a parabolic subgroup of G and P_u its unipotent radical. We consider the adjoint action of P on the Lie algebra _u of P_u. Each higher term _u^(l) of the descending central series of _u is stable under this action. For classical G all instances when P acts on _u^(l) with a finite number of orbits were determined in [9], [10], [3] and [4]. In this note we extend these results to groups of type F₄ and E₆. Moreover, when P acts on _u^(l) with an infinite number of orbits, we determine whether P still acts with a dense orbit. For G of type E₇ and E₈ we investigate only the case of a Borel subgroup.We present a complete classification of all instances when _u^(l) is a prehomogeneous space for a Borel subgroup B of a reductive algebraic group for any l ≥ 0.     

Remarks on the length of conjugacy classes of finite groups

Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by several authors. In this paper we study restrictions on the structure of a finite group in which the lengths of conjugacy classes are not divisible by p ² for some prime p. We generalise and provide simplified proofs for some earlier results.     

The Erd?s-Ginzburg-Ziv theorem for dihedral groups

Let n≥23 be an integer and let D_2n be the dihedral group of order 2n. It is proved that, if g₁,g₂,…,g_3n is a sequence of 3n elements in D_2n, then there exist 2n distinct indices i₁,i₂,…,i_2n such that g_i1g_i2?g_i2n=1. This result is a sharpening of the famous Erd?s-Ginzburg-Ziv theorem for G=D_2n.     

The coexponent of a regular p -group

A sharp bound is derived for the nilpotency class of a regular p-group in terms of its coexponent, and is used to show that the number of groups of order pⁿ with a given fixed coexponent, is independent of n, for p and n sufficiently large. Explicit formulae are calculated in the case of coexponent 3.     

Divisibility in O-primitive lattice-ordered permutation groups

We show that divisibility fails in a wide spectrum of periodic lattice-ordered permutation groups. The methods are applied to the six previously known ‘pathological’ doubly transitive groups represented on ℝ or on ℝ⁺, and only two of these groups turn out to be divisible.     

On cohomologies and extensions of cyclic groups

Cohomology groups Hs(Zn,Zm) are studied to describe all groups up to isomorphism which are (central) extensions of the cyclic group Zn by the Zn-module Zm. Further, for each such a group the number of non-equivalent extensions is determined.     

Multiplicity-free representations of symmetric groups

Building on work of Saxl, we classify the multiplicity-free permutation characters of all symmetric groups of degree 66 or more. A corollary is a complete list of the irreducible characters of symmetric groups (again of degree 66 or more) which may appear in a multiplicity-free permutation representation. The multiplicity-free characters in a related family of monomial characters are also classified. We end by investigating a consequence of these results for Specht filtrations of permutation modules defined over fields of prime characteristic.     

On self-normality and abnormality in the alternating groups

It is known that in a finite solvable group G, a subgroup H is abnormal if and only if every subgroup of G containing H is self-normalizing in G. Although, in general, the assumption of solvability cannot be dropped, in this paper we prove the theorem for the special case G = A_n and H a second maximal intransitive subgroup of A_n.Received: 1 July 2003     

On groups satisfying the minimal condition on certain subgroups

Periodica Mathematica Hungarica -     

On the Navarro-Willems conjecture for blocks of finite groups

We prove that a set of characters of a finite group can only be the set of characters for principal blocks of the group at two different primes when the primes do not divide the group order. This confirms a conjecture of Navarro and Willems in the case of principal blocks.     

On second maximal subgroups of Sylow subgroups of finite groups

Finite groups in which the second maximal subgroups of the Sylow p-subgroups, p a fixed prime, cover or avoid the chief factors of some of its chief series are completely classified.